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NONUNIFORM SPACETIME CODES FOR LAYERED SOURCE CODING By WING HIN WONG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 Copyright 2003 by Wing Hin Wong To my family. ACKNOWLEDGMENTS First I have to thank my advisor, Dr. Erik G. Larsson. Without his continu ous guidance, this thesis could never have been a reality. I am so grateful that I was given the opportunity to work on such a challenging topic. Of course Dr. John M. Shea and Dr. Tan F. Wong cannot be omitted here. I wish to express my sincere thanks to them for their supporting roles in my committee. The continuous support from my family is the key component to the success of my graduate study. The gratefulness in my mind cannot be expressed by simple languages. Last but not least, I wish to i thank you to all my lovely friends in Gainesville, Florida. Their encouragement has helped make this thesis better. TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................ iv LIST OF FIGURES ................... ........ vii ABSTRACT ...................... ........... ix CHAPTER 1 INTRODUCTION ............................. 1 2 BROADCAST CHANNELS .......... .............. 5 3 LAYERED SOURCE CODING ......... ............. 6 4 SPACETIME CODING .......................... 9 4.1 Error Performance of STBC Systems ............... 9 4.2 Differential Modulation for MIMO Systems ............ 18 4.3 Nonuniform SpaceTime Codes ....... .......... 21 4.3.1 Design Criteria ......... ..... ... ....... 22 4.3.2 Union Bound on the Error Probability .......... 23 4.4 Design Examples .................. .. .... .. 24 4.4.1 Rb = 1, R = 1 Code for 2 TX Antennas ... 24 4.4.2 Rb = 2, Ra = 1 Code ........... ... .. 30 4.4.3 Receive Diversity . ..... .... 30 4.5 Suboptimal Detector for Nonuniform DSTBC ... 32 4.6 Comparison of Differential Group Codes and A Differential Alamouti Code .... ........... ..... .. 34 4.6.1 Differential Alamouti Code . .... 34 4.6.2 Alamouti Code with Differentially Encoded Symbols 36 5 CONVOLUTIONAL PRECODING ..... . 40 5.1 Introduction to Convolutional Codes . 40 5.2 Convolutional Codes Applied to SpaceTime Coding System .42 5.2.1 HardDecision Decoding .................. .. 43 5.2.2 SoftDecision Decoding .................. .. 43 6 A NETWORK APPLICATION EXAMPLE . ..... 48 7 CONCLUDING REMARKS AND FUTURE WORKS .......... 55 REFERENCES ........... ...... ......... ...... 57 BIOGRAPHICAL SKETCH ................... ...... 60 LIST OF FIGURES Figure page 11 A pointtopoint link. .................. .... 1 12 A pointtomultipoint link .............. .. 2 31 A nonuniform 8PSK constellation obtained from a standard "uni form" QPSK constellation by splitting each original constellation point "o" into a pair of new points "x" ... .......... 8 41 Received signal level may fluctuate vastly for a single radio channel. 10 42 Total channel gain of two independent fading channels. ...... .11 43 A wireless link with multi transmit and receive antennas. ....... ..12 44 OSTBC decouples a MIMO channel into n, number of AWGN channels. .................. ............ ..19 45 Optimal values of (A, 7) for the Rb 1 R = 1 code, for a given acceptable performance degradation of the basic message. The results are obtained via minimization of the union bound (4.36) and a corresponding expression for the error rate of the addi tional message. .................. ..... 27 46 Performance degradation for the basic message and performance gain for the additional message for the Rb = R = 1 code. The results are obtained via (4.36), along with a corresponding expression for the additional message. The curves for the basic message are normalized relative to the "undisturbed < i (A 7 0) and the curves for the additional message are normalized relative to A = 0.2, 7 0.035. ................. 28 47 Empirical BER for the Rb 1, Ra = 1 code with A = 0.2, 7 0.035. 29 48 Empirical BER for the Rb = 2, Ra = 1 nonuniform spacetime constellation. The parameters were A = 0.078, 7 = 0.018 (found via optimization of (4.36)). .................. ... 31 49 MonteCarlo Simulation of the BER performance for the subopti mal receiver (4.49) and (4.50) in Section 4.5, for the joint ML receiver and for the basic message only system with Rb = 2, R, = 1 code. The curves " x " and " x " overlap. .... 38 410 Monte Carlo Simulation result for the BER performance of Rb 2, Ra = 1 system, with differential group codes and differential Alamouti codes. Note that the solid and dash curves with "x" marks overlap. .................. ..... 39 51 A simple rate 1/2 convolutional encoder. . ...... 41 52 With rate1/2 convolutional coding, comparison of BER perfor mance are shown for harddecision detection, softdecision detec tion with concentrated likelihood function, and softdecision with estimated noise variance. .................. .... 47 61 7cell frequency reuse system. .................. .... 49 62 Typical cell layout for a wireless telecommunication system. 50 63 Coverage area for the basic/additional messages as a function of the "tolerable" performance loss, for the Rb = R = 1 code. The horizontal curves represent the relative coverage area for the Rb = 2 code with a basic message only. .............. ..53 64 Coverage area for the SISO basic/additional messages as a function of the "tolerable" performance loss. The horizontal curves rep resent the relative coverage area for the SISO rate2 code with a basic message only. ............... .... .. 54 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NONUNIFORM SPACETIME CODES FOR LAYERED SOURCE CODING By Wing Hin Wong May 2003 C('! In: Erik G. Larsson Major Department: Electrical and Computer Engineering We propose new spacetime codes tailored to pointtomultipoint, or broadcast, communications using 1 ,,. i t source coding. Our codes can be encoded (and decoded) differentially, and they are based entirely on phaseshift keying. We develop analytical design criteria for the codes, and we discuss the design of optimal and suboptimal receiver structures. We also discuss the relation between our new codes and a differentially encoded Alamouti code. Numerical examples illustrate the performance of our new codes. Convolutional codes are also introduced as a precoding to the spacetime coding. With some reduction of the data rate, we can significantly increase the error performance in an adverse channel environment. Softdecision decoding is being used to further improve the performance of the convolutional codes. For completeness, we also include some fundamental concepts of the spacetime coding system and the cellular system for the ease of reading for newcomers in this field. CHAPTER 1 INTRODUCTION During the last decade, wireless systems with multiple transmit and receive antennas have been studied extensively, and the performance of such systems has been proven to be extremely promising. The propagation channel associated with a system with multiple transmit and receive antennas is sometimes called a multipleinput multipleoutput (\! M0O) channel, and the associated coding and signal processing is referred to as spacetime coding. Using proper spacetime coding, it is possible to use the degrees of freedom of the MIMO channel both to increase the throughput and to counteract fading. For this reason, MIMO technology is believed to become a ii' Pr cornerstone in many future wireless communication systems. Loosely speaking, communication links can be classified into two categories: pointtopoint links and pointtomultipoint links. In the former case, there is exactly one transmitter and one receiver which communicate with each other at a given time, whereas for a pointtomultipoint link, a transmitted message is aimed at multiple different recipients simultaneously. A cellular communication system with mobile users is an example of a pointtopoint communication Figure 11: A pointtopoint link. Figure 12: A pointtomultipoint link. system, whereas a radio/TV broadcast is an example of a pointtomultipoint link (therefore pointtomultipoint links are sometimes referred to as broadcast channels). Pointtomultipoint links are becoming increasingly important. For in stance, the introduction of Digital Audio Broadcast (DAB) and HighDefinition Television (HDTV) has pioneered a whole new field of digital broadcasting applications. As a further example, it is widely believed that much of the next generation's wireless networking will be based on socalled adhoc networks, where it may be necessary for multiple units to listen to one message at the same time. Also, in conventional cell communication system using directional or adaptive antennas, it is sometimes necessary to broadcast a message to the entire cell. There are two 1i ii differences between pointtopoint and pointto multipoint communication links. First, a pointtopoint connection can be optimized for a given transmitterreceiver pair. For instance, a cellular system usually employs power control techniques that adjust the transmitted power to minimize the power consumption, reduce the amount of cochannel interfer ence, and at the same time ensure that the received signal strength exceeds a certain threshold. Second, in contrast to a pointtopoint communication link, for a broadcast transmission all receivers have different capabilities to decode the transmitted message. This is so because the different receivers experience in gen eral very different radio link qualities (for instance, due to varying propagation conditions). Moreover, since quality can usually be traded for cost, the receivers themselves may have different inherent abilities to decode the transmitted message. The fact that distinct receivers have different capabilities of decoding a mes sage si l. I that the transmitted signal should consist of several components which are of different importance for the reconstruction of the message (and therefore have an inherently different vulnerability to transmission errors). This idea has lead to the concept of li, . It source coding (e.g., [1, 2, 3, 4]), which is now a mature technique employ, l1 in many multimedia standards. For instance, the image coding standard JPEG2000 and the video coding standard MPEG4 use what is sometimes referred to as "fine granularity scalability," which enables a gradual tradeoff between the errorfree data throughput and the quality of the reconstructed image or video sequence [5]. Such progressive source coding methods are already in use in many Internet applications where data rate can be traded for quality, and they are expected to pl i, an instrumental role for the next generation of wireless standards to provide ubiquitous access both to the Internet, and to diverse sources of streaming video and audio. The topic of this thesis is spacetime coding for broadcast channels when 1i .. I1 source coding is used. Previous works on the topic include Memarzadeh et al. [6], which discussed "b( ,inl,,i ,iinii, techniques (hence requiring feedback of channel state information) for a multiuser system. In Memarzadeh et al. [6], and two kinds of beamforming techniques are discussed. Zeroforcing beamforming is introduced which orthogonal codewords are used for each user so that interuser interference can be fully eliminated just as in an ideal CDMA system, provided that exact channel state information is available at the transmitter. Another way is the single user optimal beamformer, for which no attempt is made to avoid interuser interference. This is suitable for the scenario when the channel state information at the transmitter is not very accurate, as interference suppression is not effective. Kuo et al. [7] proposed a 1i,. ir spacetime coding scheme assuming different receivers may have different numbers of receive antennas, and that a receiver with more receive antennas can decode more 1 i vrs of messages. In this thesis, we sir.. I a new l t1 r, spacetime coding scheme that does not require any channel knowledge at the transmitter, and which is constructed starting from a criterion that attempts to minimize the error rate. Our new codes also satisfy two other important design goals. First, they can be encoded (and detected) differentially. Second, they are entirely based on phaseshift keying (PSK) modulation and consequently the transmitted signal has constant envelope at all times. The codes proposed in this paper can therefore be seen as a multidimensional extension of a transmission technique for singleantenna systems in Pursley et al. [8]. CHAPTER 2 BROADCAST CHANNELS Broadcast channels are defined as the simultaneous communications between a single source and several receivers. Early works include Cover et al. [9] and Bergmans et al. [10]. In Cover et al. [9], They discuss the nature of broadcast channel from the perspective of information theory, and derives upper bounds of the capacity of broadcast channels of different types. It is proved that the maximum data rate of each channels are jointly achievable by proper channel coding. By superimposing side information to the common message intended for all receivers, different data rates are possible for different channels. This idea is similar to the nonuniform codes which we will discuss later on. Due to the different capabilities of each receivers, and indeed the capabilities of each receivers are varying from time to time (i.e., due to channel fading or fluctuating noise level), the theoretical maximum capacity of a broadcast channel is not easy to achieve in a practical system. We have no intention to compare the performance of our nonuniform codes with the theoretical channel capacity in this thesis, but as our nonuniform codes contain data in multiple l V,i we believe that we are in the right direction that our nonuniform codes can increase the total throughput of a broadcast system. CHAPTER 3 LAYERED SOURCE CODING In this chapter we briefly introduce the concept of 1 it, I source coding. JPEG2000 is taken as the example here. JPEG is an image compression standard developed by the Joint Photo graphic Experts Group [11]. For the encoding, the pixel bit map information of a whole image is divided into 8 x 8 pixel blocks. Each 8 x 8 pixel block then undergoes a discrete cosine transformation (DCT). This helps separate the image into parts (or spectral subbands) of different importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform as it transforms a signal or image from the spatial domain to the frequency domain. For most images, much of the signal energy lies at low frequencies, whereas the higher frequency components are often small and usually they can be neglected with little visible distortion. Thus the DCT indeed separates the original image data into separate lv. i~ of different importance: the low frequency components are important for the reconstruction of the image; the higher frequency terms are not that important although they enhance the visual quality of the image. From this observation, we infer that the low frequency parts of data must be sent through a reliable data channel with very good biterrorrate (BER) performance whereas the high frequency parts can be sent through a less reliable channel as the occurrence of bit errors will only deteriorate the quality to a negligible extent. With 1 l,. i1. t source coding (for instance as mentioned above, JPEG2000 and MPEG4), more capable receivers, for example with higher signaltonoise ratio (SNR), can achieve a higher data rate by decoding the messages contained in all li.i~ while the less capable receivers can only decode the message in the bottom 1', r. Thus the i, .I important" information should be coril., , '1 via the bottom 1'.r while "less impe I 1,il information can be transmitted via higher lv,r (s). An example of a modulation scheme designed for 1 li, 1t 1 source coding is nonuniform phaseshiftkeying modulation (PSK), which was first introduced by Pursley and Shea [8] (see also Pursley et al. [12, 13]). The idea is intuitively appealing. Starting with a standard uniform PSK constellation, which contains data for the I/'i..." 1i vr, a nonuniform PSK code constellation is obtained by adding a small additional phase shift a to each original constellation point, which contains the information for the "additional" lVvr. This is illustrated in Figure 31, when the basic message is con,:.', i, 1 via a QPSK constellation. A capable receiver with high SNR can distinguish among all eight constellation points which means that both the basic and the additional 1,vr messages can be decoded, whereas to a less capable receiver the constellation may appear like a blurred QPSK constellation and hence it may only be able to distinguish between the different "1 i;,, phase shifts; thus only the basic message can be detected accurately. By such a construction the error rates associated with the basic and the additional message are different. Consequently, the additional message can carry information that is of less importance for the reconstruction of the transmitted message than the basic message is. The error probability of the additional message can be easily adjusted by choosing different values of the additional phase shift a. Obviously, a larger a results in a better performance x x U xTx Figure 31: A nonuniform 8PSK constellation obtained from a standard "uni form" QPSK constellation by splitting each original constellation point "o" into a pair of new points "x". for the additional message at the cost of a deterioration of the performance for the basic message. The main virtue of this nonuniform PSK encoding scheme is that the signaling envelope is constant while multii1 i, t messages can be transmitted with different and adjustable error performance. CHAPTER 4 SPACETIME CODING 4.1 Error Performance of STBC Systems A conventional wireless system with a single transmit and receive antenna suffers from a large performance degradation due to the fading of the radio channel. Fading is the fluctuation in received signal level. It is mainly due to changes in the propagation environment, such as weather and moving reflective objects/obstructions. Fading can be classified as fast fading or slow fading, depending on how rapidly the channel impulse response changes. For a fast fading channel, the channel impulse response changes rapidly within one symbol duration, which means that the coherence time of the channel T~ (which is a measure of the time duration that the channel impulse response is quasistatic) is smaller than the symbol period of the transmitted signal T, > T (4.1) For a slow fading channel, the channel impulse response changes very slowly compared with one symbol duration, which means that the channel can be seen as quasistatic for many symbol durations, that is T, < T, (4.2) Figure 41 shows an example of the fast fading of the gain of a single radio channel. The two curves represent two channels encounter independent fast fading. We can see that there is a high risk that a single channel may encounter 25 i 20 : 10 5 i' 0 5 101 Figure 4 1: Received signal level may ! ,,.:tuate vastly for a single radio channel. a deep fade at a certain time. During deep fade the transmission quality may be very poor or the transmission may simply be impossible. To counteract the fading radio environment a multiple transmit and receive antenna system (or MIMO) is one attractive possibility. The advantage of a multiple transmit and receive antenna is not difficult to comprehend intuitively: There are more than one channel which are not fully dependent, so the possi bility that all of them encounter a deep fading at the same time is small; thus the deep fading problem may be solved. Figure 42 shows the total channel gain (denote the individual channel gains be hi and h2, total channel gain is then equal to ih1 2 + lhl2) of two independent fading channels. It can be seen that the risk of the total channel gain encountering a deep fade is largely reduced. A  :: {' :iZ.,i: ". 2000 S10  S5 0) c 5' 5 10 15 20 I I 0 500 1000 1500 2000 time Figure 4 2: Total channel gain of two independent fading channels. The corresponding encoding scheme for such a multi transmit and receive antenna system is called spacetime coding. This is an emerging topic that now attracts many researchers' attention since its introduction about a decade ago. The main virtue of such a multi transmit and receive antenna system is that higher diversity order can be achieved. The diversity order can be defined implicitly by BitErrorRate oc (SNR)diversity (4.3) which means that the biterrorrate decreases as fast as the SignaltoNoiseRatio (SNR) to the power of minus the diversity order. For a high SNR the advantage of a high diversity order can be substantial. Consider a MIMO system with nt transmit antennas, n, receive antennas (see Figure 43), and assume for simplicity that the propagation channel is TX Y j/Y I Y Y nt nr Figure 4 3: A wireless link with multi transmit and receive antennas. linear, timeinvariant and frequency flat. Let H be an n, x nt matrix whose (m, n)th element contains the channel gain between transmit antenna n and receive antenna m, and suppose that a code matrix X of dimension nt x N is taken from a matrix constellation X and transmitted by sending its N columns via the nt antennas during N time epochs. If the signal received during the same N time intervals is arranged in an n, x N matrix Y, then Y =HX + E (4.4) where E is an n, x N matrix of noise. Throughout this thesis, we shall make the somewhat standard assumption that the elements of H and E are independent and zeromean complex Gaussian with variances p2 and a2, respectively; hence the channel is Rayleigh fading. The Shannon channel capacity of such a MIMO system is a wellknown result, and it can be shown that (for details see [14] chapter 3 and the references therein) C(H) = B log2 II + HPHHI (4.5) where B is the bandwidth of the channel in Hz and P is the covariance matrix of the sequence of transmitted data vectors P = E[xnxH] (4.6) If the transmitted signals are uncorrelated at different transmit antennas and we use unit power at each transmit interval, then 1 P I (4.7) (4.5) reduced to C(H) B log2 I + HHHI (4.8) ntJ2 Clearly, the channel capacity depends on the eigenvalues of HHH. Given Y, detection of X in a maximumlikelihood (\I ) sense amounts to minimizing the Euclidian distance IY HX11 (4.9) with respect to X e X, as well as H (unless it is known). The average (over H) probability that the ML detector mistakes a transmitted code matrix Xo for an incorrect (and different) code matrix X / Xo is called the pairwise error probability For completeness a brief derivation of this probability is provided here for both coherent and noncoherent detection. Coherent: Suppose that Xo is the true transmitted code matrix, thus Y = HXo + E. Then the probability that an incorrect decision in favor of a different code matrix X / X0 occurs is P(errorIH) =P(IIY HX2 < IIY HXo\ 2 H) =P(IHX + El2 < IIE 12 H) =P(2ReTr{EHHX} > IHX 2 H) (4.0) where X = Xo X, and I denotes the determinant of a matrix. In (4.10), the second equality follows by some matrix algebra and the last step is a conse quence of the fact that 2 Re Tr{E*HX} is zeromean Gaussian with variance 2j2 H HX 12 together with the ('! i 11i, bound. Under the assumptions made, the probability density function (p.d.f.) for H is p(H) =exp 2) (4.11) fnrfntp2nrnt 2 Averaging (4.10) over H using (4.11), we get after some calculations P(error) dH P(errorIH)p(H) < (Tp2) t dH exp ( 1 HX2 4H 2) (4.12) S(.p2) nrt dH exp( Tr H ( XXH + I) H p2 ~ H nI Tr < I2(X P "rnt = XH+ I < (X Xo)(X Xo)H I nr 4j2 472 Noncoherent: If H is unknown, we can obtain a detection rule that depends only on X. Consider the problem of minimizing (4.9) min IY HX 2 (4.13) XEX,H 15 Treating H as a deterministic unknown, we can minimize this likelihood function with respect to H. Notice that IY HX I2 y_ H XHHH 2 (xH + H )(YH XHHH) 2 = IIx(y XHHH) 2 H(YH XHHH 2 41 +2 Re (Tr {(Y HX)HIx HH (YH XHHH) }) nX YH XHHH 12 + H n_,y 2 Here IHxH XH(XXH)1X is the orthogonal projector of XH. The function in (4.14) is minimized when HXHYH XHHH = 0 ==XH(XXH)1XyH XHHH = 0 (4.15) tXH((XXH)XYH HH) = 0 If X has full rank then the above leads to the result (see, e.g., [15] and [14] appendix B) H =YXH(XXH)1 (4.16) (It may be argued that if we know some sort of statistical properties of H, we should treat H as a stochastic variable when it enters the likelihood function. This is intuitive since we should incorporate all known information into the detector to get the best result. But in Larsson et al. [16] (see also Larsson et al. [14] exercise 9.7) it is shown that both approaches indeed give the same result. In this paper we will stick to the deterministic channel approach for simplicity.) Inserting (4.16) into (4.13) yields min IY YXH(XXH)1Xl2 XEX mi mmin lYrll2 (4.17) XEX Smax IILxyH 112 XEX Let An xf nXH (4.18) Note that the matrix An is Hermitian, but not necessarily positive definite. Also note that AnXH = (fXH _HXH)XH = X H nHXOH (4.19) 0 0 IIxHXOH Hence, since IIxHXoH has full rank by assumption, the matrix XolixHX0H is positive definite. The probability that the ML detector makes a mistake is P(error H) P P(IXYH I2> IflXHyH 2 H) P(Tr {(HXo +E )A(HXo +E )H} < H) =P(2ReTr {HXoAnEH} (4.20) + Tr{EAnEH} < Tr{HXoAnXHH} H) =P(2ReTr {HXoIx HEH} + Tr{EAnEH} < IlIHXoHH 2 H) X 0 where in the last step we used the fact that Hl XlHL = nH . 17 The term EAnEH is of higher order and may be neglected in an .imptotic analysis. In particular, for a given H, the variance of its elements is of the order a4, whereas the elements of HXollUHEH have a variance of the order a2. Since dQ(x) 1 x2 1 S2 < < o (4.21) dx 2w Thus a small finite error in variance can only caused a small finite error in the Qfunction also, so the additional small variance caused by this higher order term can be neglected. By the C'!I. 1 r !' bound we have P(2ReTr{HXon HEH< Iln XHX H 2 H i( IHXOH 2 (4.22) < exp ( 42 ) Hence, by averaging in a way similar to (4.23), we get EH[P(error)] I dH P(errorH)p(H) < (TP2 ) exp I xHH H 1 11H11 SdH e 42 H 2(4.23) S(i~p2),rnt dH exp ( Tr H. ( Xo HXH + I) H 4 Xox HX o + < IXollHXo I ,4) f To summarize, the error bounds for coherent and noncoherent detection in a STBC system are the following E [P(Xo X)] c < (X Xo)(X Xo)H nr (42) t (4.24) coherent 4j and E [P(Xo X)] < Xon Xo n (p) (4.25) noncoherent 4 2 A special subclass of spacetime codes is the linear or/',. ',. ','l spacetime block codes (OSTBC). The case when OSTBC applied to a 2TX system turns out to be the wellknown Alamouti code [17]. The main virtue of OSTBC is that they achieve a diversity of order nnt for a system with nt transmit and n, receive antennas, at an almost negligible com putational cost. Specifically, for such a system, a set of n, symbols {Si,..., s,} are encoded into an nt x N matrix that has the following structure nts XOSTBC (A, + s B,) (4.26) n=l The columns of XOSTBC are transmitted via the nt transmit antennas during N time intervals. In (4.26), {AT,B,} are matrices chosen such that XOSTBCXOSTBC C 12. I (4.27) n=l for all {s,}. In general, the role of the orthogonality condition (4.27) and the associated design of {AT, B,} is relatively well understood by now; for instance, (4.27) implies that the symbols {s,} can be detected independently of each other [14], so with the use of OSTBC, the MIMO channel can be decoupled into n, number of single additivewhitegaussiannoise (AWGN) channels (See Figure 44). 4.2 Differential Modulation for MIMO Systems Spacetime modulation matrices that can be encoded differ. ,:l.,iall are sometimes of special interest since such codes can easily be demodulated nonco herently. Differential codes with uniform error probabilities have been studied by many authors (see, e.g., [18, 19, 20, 21, 22] for some prominent examples) and some of them can be seen as an extension of differential PSK to MIMO systems. In general, if t is the time index and if {U(t)} is a sequence of (square) 19 AW Figure 44: 0' :lC d1co( .1 a MIMO0 channel into 1 number <.fl "C XWN chan lsignal  1el t . informationbearing matrices, then differential encoding obtains the transmitted code matrix X(t) at time t via X(t) X(t )U(t) (4.28) where X(t 1) is the matrix transmitted at time t 1. Such an encoding usually is only meaningful under certain circumstances, for instance if U(t) is unitary (in which case X(t) becomes unitary for all t as well, given a unitary "initial iii i::: X (0)). By considering two (or more) received matrices Y(t) and Y(t 1) simultaneously, noncoherent detection is possible. For example, if we concatenate the matrices received at time t 1 and t, and assume that the channel H remains constant over these two time intervals (to within practical accuracy), we can write (using (4.4) and (4.28)) Y(t ) Y(t) HX(t 1) HX(t) + E(t 1) E(t) (4.29) =HX(t 1) HX(t )U(t) + E(t 1) E(t) HX(t ) I U(t) + E(t ) E(t)I where HX(t 1) can be seen as an unknown effectivee channel matrix and rI U(t)j is an effective code matrix. Provided that rI U(t)j is such that the corresponding determinant in (4.25) is nonzero, noncoherent demodulation is possible. By applying the decision rule (4.17) to the data model (4.29), the detection rule for differential demodulation becomes 2 min (t) Y(t ) Y(t) I s] Ymin U(t)EX 1 U11  ^ mmin Tr y(t 1) Y(t) u(t)x TrY 1) Y (t) u ) *^ f ^^ f ^ ^ J ^ .^H\ ~ H(t 1) Y(t)] 1(t) YH a max Re Tr U(t)Y" (t)Y (t 1) U(t)EX where Tr{} stands for the trace of a matrix and where we used the fact that the informationbearing matrix U(t) is unitary. The associated average error probability, i.e., the probability that U(O) is incorrectly detected as U for this differential detection scheme can be bounded by using (4.25) I r<[rt E [P(U( o) U )] < U ( o) H 2. ( ) () U()H 4(4 .31) 1 r" I (UU()H +Uu H 2 S2 )nt 4(T2 For simplicity hereafter the time index t of U(t) is omitted whenever no confu sion can occur. 4.3 Nonuniform SpaceTime Codes Inspired by the nonuniform PSK codes discussed in Section 3, our new nonuniform spacetime codes are based on differential encoding of the product of a unitary code matrix Ub E Xb associated with a basic message, and another unitary code matrix Ua E X. corresponding to an additional message. Thus the transmitted code matrix at time t is given by (cf. (4.28)) X(t) = X(t 1)Ub(tt)U t) (4.30) (4.32) Using (4.31) along with the fact that UbUa is unitary, we find that the average probability that a transmitted message pair (U ,Uo) is mistaken for another pair (Ub,Ua) can be bounded by E [P(Uo, U Ub, U)] SI (UbUaUOHUH + uuuru ) 2 ) nrnt (4.33) A bound such as (4.31) or (4.33) was used in, for instance, Hughes et al. [18] for the design of (uniform) differential spacetime codes. However, although it may be thought of as a feasible approach, an attempt to minimize this bound in the context of r7. ,;r. :. rm spacetime modulation may not produce the desired result since the target error rates for Ub and Ua are different. 4.3.1 Design Criteria Since the additional message can be decoded only at high SNR, the matrices {Ua} associated with the additional message should approximately be close to the identity matrix. Inspired by this observation, we can first consider the design of {Ub}, treating the presence of Ua as an unmodelled noiselike disturbance term. Doing so, we can approximately bound the error probability for Ub alone (assuming differential detection) by E [P(U Ub)] I (u(0) bH + UbU0()H) n ) 2t (4.34)2  where a2 is a factor that incorporates the noiselike effect of the presence of Ua. Although the "bound" (4.34) is somewhat heuristic (and probably neither tight nor very accurate), we believe that it may serve a purpose as a meaningful design criterion for {Ub}. Next, for the design of {Ua} we proceed as follows. Suppose that the SNR is in a region such that Ub can be reliably decoded. For the design of {Ua} this should be a reasonable assumption, since if it is not true then decoding of U. is probably of less interest anyway. Assuming that Ub is known, the demodulation of Us is essentially another noncoherent detection problem. To obtain a criteria for the design of the constellation {Ua}, we want to form an error bound on U. and average it over all possible basic messages {Ub}. Using the bound (4.25) for noncoherent detection appears to yield criteria that are very difficult to use. As a suboptimal approach we used instead the bound (4.24) for coherent detection. Doing so results in the following criterion E [P(U Us)] < (U UbUa)(Ub U UbU)O r 2 H " Ub EX (U U)(U Ua)H n. ( 2) nt (4.35) where in the last step Ub disappears since it is unitary. In (4.35),  denotes the number of elements of the set. 4.3.2 Union Bound on the Error Probability By the union bound, the probability for the basic message to be in error (averaged over all possible pairs of additional messages) can be bounded by E [P(basic message in error)] 1 1 (U)b ,U[))CXb (Ua Ua )EXa k4n I (U( k)U(r)U()HU b)H + U)u))U(r)HU (k)Hk ) nt (4.36) Likewise, the error rate for the additional message (averaged over all possible pairs of basic messages), can be bounded by a similar expression E [P(additional message in error)] 1 2 I (Uk) U(U)HUn)H + U ( us) U()HUk)H nrnt (4.37) These bounds will be useful for performance optimization. 4.4 Design Examples 4.4.1 Rb R = 1 Code for 2 TX Antennas We first construct a code where the rate for the basic message is Rb 1 bit/sec/Hz and the rate for the additional message is Ra = 1 bit/sec/Hz as well. We take the basic message Ub taken from the following set 1 0 0 1 0 1 1 0 SX= (4.38) 0 1 1 0 1 0 0 1 The constellation in (4.38), which is uniform and possesses certain optimality properties, is due to Hughes et al. [18] and was called "BPSK" therein. Based on the design rules in Section 4.3.1, we have handcrafted the following constellation of matrices for the additional message Ua ( (i ear 0 e0e 0 eb e 0(o Xa { } (4.39) S0 ea 0 e"t 0 ed 0 e0 o where (A, 7) are design parameters (to be discussed below). The constellation may appear to be somewhat arbitrary but it possesses some nice features. First, all matrices in (4.39) are unitary, this is a requirement for MIMO differential encoding. Second, it is symmetric and uniform which means that the error probability performance should be the same for all constel lation points. Also similarly to the nonuniform PSK encoding for a single channel (cf. Section ??), multiplication with the matrices in (4.39) can be interpreted as adding a small phase shift to the elements of the basic message Ub. We may also think of other forms of matrices as the additional message. For instance, consider the following antidiagonal matrix 0 eix Ua = (4.40) But we immediately find that this does not work. Consider, for example, 1 0 Ub (4.41) 0 1 Then, 1 0 0 e" 0 e"^ Ub Ua = (4.42) 01 et O e07 0 It is seen that multiplying this additional message Ua not only introduces a small phase shift but also interchanges the columns of the original basic message Ub. This is not acceptable since our design goal is to have the additional message only pose a small disturbance to the basic message so that for those lesscapable receivers can still detect the basic message even without knowledge of the existence of the additional message. The U. here simply destroys the original basic message constellation so it is not feasible. Let us now return to our design task. If we take the initial transmit matrix, somewhat arbitrarily X(0) 1 (4.43) The factor is inserted to normalize the transmitted power to unity for each time interval. It follows that XH(t)X(t) I= (4.44) for all t and hence the differential encoding of the code is meaningful. Further more, we can verify that all elements of X have ahv, constant magnitude: Xk,l(t) = (4.45) and hence all transmitted symbols are obtained by constant envelope modulation, which was one of our design goals. The error performance (of both the basic and the additional message) will depend on A and 7, and typically the error performance of the additional message can be traded against that of the basic message. The values (A,7) must be chosen carefully. For instance, at first glance intuition would perhaps sir. 1 us to take A > 0 and 7 0, but we can show that such a wiIl,!." choice leads to a code that v.1i I:' but that does not provide maximal diversity. In our experiments, we used the union bound (4.36) and (4.37) to optimize over (A, 7). In particular, for a given "tolerable" loss in performance for the basic message, we can find the pair (A, 7) that minimizes (4.37). The result is shown in Figure 45. For example, if we can accept a degradation of 1.5 dB for the basic message, then A = 0.2 and 7 = 0.035 are optimal. The optimization over (A, 7) is 0 0.5 1 1.5 2 2.5 3 3.5 4 Maximum tolerable performance loss for basic message (dB) Figure 45: Optimal values of (A, 7) for the Rb 1 R = 1 code, for a given ac ceptable performance degradation of the basic message. The results are obtained via minimization of the union bound (4.36) and a corresponding expression for the error rate of the additional message. further illustrated in Figure 46, where we show how the performance associated with the basic and additional messages varies with (A, 7). Figure 47 shows the empirical biterrorrate (BER), obtained via Monte Carlo simulation, for the code described above using A = 0.2, 7 = 0.035 and ML decoding. The solid lines ("") show the performance of differential nonuniform BPSK for a conventional system with it = 1 transmit antenna and a single receive antenna (this is essentially a special case of Pursley et al. [8]). The dashed lines (" ") show the performance for a system with nt = 2 (and a single receive antenna) using the new code presented above. For the curves without marks, only a basic message is transmitted. The curves with marks show the performance when both a basic and an additional message are transmitted: the 60 50 40 ',I 5 0 30 0 5 20 t 10a 0.05 .05 10 0 0.1 0.05 0 .1 5 0.15 0.2 0.2 0.25 0.25 Y Figure 46: Performance degradation for the basic message and performance gain for the additional message for the Rb = R = 1 code. The results are obtained via (4.36), along with a corresponding expression for the additional message. The curves for the basic message are normalized relative to the "undisturbed < i (A = 7 = 0) and the curves for the additional message are normalized relative to A 0.2, 7 0.035. curves marked with "o" show the BER for the basic message, and the curves with "x" show the BER for the additional message. Clearly, the transmit diversity system outperforms the conventional one observe, in particular, the different slopes of the BER curves both for the basic and for the additional message. The simulation also confirms that the transmission of an additional message incurs a small performance degradation for the basic message. 29 10  nt=l, DBPSK (basic msg. alone)  nt=1, DBPSK (basic msg.) nt=1, DBPSK (add. msg.) n2 t=2, Code (4.38) (basic msg. alone) 10 nt=2, Code (4.38)(4.39) (basic msg.) (, , nt=2, Code (4.38)(4.39) (add. msg.) $10 S  I t m *% 104 105 15 20 25 30 35 Signaltonoise ratio [dB] Figure 47: Empirical BER for the Rb 1, R = 1 code with A 0.2, 7 0.035. 4.4.2 Rb = 2, R = 1 Code To obtain a code with a higher information rate for the basic message, we next take Ub from the following algebraic group of 16 matrices (which is due to Hughes et al. [18] as well) generated by ( ej,/4 0 0 1 \ b =( (4.46) 0 ej/4 1 0 The notation in (4.46) means that all possible matrices can be generated by choosing any arbitrary integers M and N in the following expression M N ejw/4 0 0 1 (4.47) 0 ej/4 1 0 The constellation used for the additional message is chosen to be the same as (4.39), so we now have a system with rate Rb = 2 for the basic message and rate Ra = 1 for the additional message. The corresponding simulated BER is shown in Figure 48. 4.4.3 Receive Diversity To demonstrate the transmit diversity achieved for the 2TX system in the simplest way, our results above are based on the 1RX case. Although of this, our newly proposed nonuniform STBC codes are not only useful for 1RX case but also for a general n,RX case. This is obvious since our theoretical derivations above are not tailored for the 1RX case, hence it should work automatically for a general n,RX case. Also we could expect a diversity order of nnt can be achieved. This is so since we can see the data model of a n,RX ntTX system as **% * K 101 102 2 ( 3 10 Il 0 I, m 103 % I 30 Signaltonoise ratio [dB] Figure 48: Empirical BER for the Rb = 2, Ra stellation. The parameters were A = 0.078, 7 = (4.36)). n, different 1RX ntTX system Y =HX+ E Y, Y, Yn = 1 nonuniform spacetime con 0.018 (found via optimization of (4.48) H, X + En, 4Q 44k 4, nt=l, DQPSK (basic msg. alone)  nt=1, DQPSK (basic msg.)  nt=1, BPSK (add. msg.) nt=2, Code (4.46) (basic msg. alone) O nt=2, Code (4.46)(4.39) (basic msg.) . nt=2, Code (4.46)(4.39) (add. msg.) H1 El X +E HX+E E, HIX + E, 0 105 2( where Yk for k =1,. n, is the 1 x N received data at the kth receive antenna, Hk is the 1 x nt channel matrix associated with the kth receive antenna, Ek is the 1 x nt noise matrix at the kth receive antenna. As usual, for simplicity if we assume the channel matrices associated with n, different receive antennas Hk (for k = 1, n,) are independent and the noise is white (El, E, are uncorrelated), then the n,RX ntTX can be broken down into n, number of independent 1RX ntTX system. Each receive antenna has sufficient data to detect the transmitted matrix X so they can work individually and achieve transmit diversity of order nt. To further utilize receive diversity we can employ joint ML detection across all n, independent receive antennas thus increasing the overall diversity by a factor of nr. Since the n, receive antennas are independent at the receiver site, so optimizing the space time encoding scheme for a single receive antenna is equivalent to optimizing a system with n, independent receive antennas. 4.5 Suboptimal Detector for Nonuniform DSTBC To detect both the basic and the additional messages in a joint ML fashion we have to search through all IXb IX. possible combinations to find the combination of Ub and Us which maximize the cost function in (4.30). This may be computationally burdensome if the constellation sizes of both lwris are large. A possible remedy to this problem is to use a suboptimal detection rule described as follows Step 1. Detect the basic message only using (4.17), neglecting the presence of the additional message r 2 ub (t)= b in Y(t Y(t)] ub(t) tUb(tEb L (4.49) Smax Re (Tr Ub(t)YH(t)y(t )} Ub (t)EXb Step 2. Detect the additional message assuming the decision taken in Step 1 is correct r 2 mm Y(t ( U t) U ) Y t) UH U 1(4.50) m> ax Re Tr U (t)YH Y l) Ua(t)EXa L I Let us elaborate more on why this suboptimal detection scheme works. Since Ua is close to the identity matrix, when compared to the basic message Ub it can be treated as small unmodelled noise; hence the detection rule in Step 1 can still be able to detect the basic message with only a small performance degradation. As the error performance of the basic message is typically superior to that of the additional message, we can assume the detected basic message is correct when we are dealing with the detection of the additional message. Thus the observation model for the detection of Ua(t) becomes Y(t 1) rb(t) Y(t)] =HX(t b() Ub(t)Ua(t) + E(t ) E(t) (4.51) HX(t 1)b(t) I Ua(t)] + E(t 1) E(t) Treating HX(t 1)Ub(t) as an unknown effectt, channel as in (4.29) and using (4.17) and (4.30) yields the detection rule in Step 2. By applying this suboptimal detection rule the computational complexity is reduced from IXb IX  to IXb + 11. For example, for the R = 2, Ra = 1 code, the computational time is reduced from XbI, = 16 4 = 64 to IXb + I1 = 16 + 4 = 20. For this example the reduction is not very significant; yet this suboptimal detection scheme does provide more flexibility as opposed to the joint ML detection because if only the basic message is needed, the computational load will be further reduced to XbI = 16 (for this example). We expect that there is some performance loss associated with this subopti mal detection scheme since it does not exploit knowledge of the structure of the additional message in the Step 1. The MonteCarlo simulation result in Figure 49 shows the performance of the suboptimal detector for the Rb = 2, Ra = 1 code compared to the performance of the joint ML detector. The detection of the basic message gives a small performance loss (a 0.3 dB) compared to the joint ML detector, whereas the detection of the additional message is (to within practical accuracy) equal to that of the joint ML detector. 4.6 Comparison of Differential Group Codes and A Differential Alamouti Code 4.6.1 Differential Alamouti Code The main advantage of the nonuniform spacetime group codes introduced in Section 4.4 is that the signal envelope is constant. However, also for this reason they cannot achieve the maximum possible performance (in terms of BER) because they do not make use of any amplitude information in the signal. On the contrary, orthogonal spacetime block codes (OSTBC) should provide better performance (BER) since they are already optimized by construction [17, 23, 14]. For 2 TX antennas OSTBC reduces to Alamouti code. The construction of an Alamouti code matrix is as follows U(t) ( ) 2 (4.52) where si and s2 are the information bearing symbols and (.)* stands for the complex conjugate. For the Rb = 1 system, si and s2 are taken from the BPSK set {1, 1}, and for the Rb = 2 system, they are taken from the QPSK set {1, , ,j}. The 1//2 factor is included for normalization purposes so that the transmitted power per time interval equals one. This Alamouti matrix U(t) can then be encoded differentially by using (4.28) in Section 4.2. Note that the Alamouti code works together with the additional message U. taken from (4.39) to form a nonuniform constellation, and differentially encoded by (4.32), but unfortunately the matrices in the soobtained constellation do not have constant signal envelope. Here we compare the performance of the group codes to that of the Alam outi code. The Rb = 1 group codes (4.38) in Section 4.4 and Rb = 1 Alamouti codes are indeed equivalent, so there is no performance advantage for Alamouti codes for this case. The equivalence of both codes is easy to show as in this case the set of Alamouti matrices become 1 1 1 1 1 1 1 1 1 1 1 1 Xa amout 7 { [ j (4.53) V 1 1 1 1 1 / 1 1_ t If we premultiply the above set by ] then we obtain the Rb 1 group code in (4.38). For the Rb = 2 codes, the Alamouti code does provide somewhat better BER performance than the group codes in (4.46) and (4.39). This is so since the Alamouti code uses both signal amplitude and phase to transmit informa tion; thus it should be less susceptible to the phase disturbance caused by the additional message. Figure 410 shows the simulated performance difference between the Rb = 2, Ra = 1 nonuniform group codes in (4.46) and (4.39) and a Rb = 2 Alamouti code together with the additional message (4.39). For the basic message the Alamouti code outperforms the group code by about 1dB, and it is somewhat more robust when additional message is added. Meanwhile the performances of the detection of the additional message are equal both when the Alamouti and the group codes (4.39) are used as the basic message carrier. 4.6.2 Alamouti Code with Differentially Encoded Symbols It may be argued that both the constant signaling envelope and maximum BER performance can be achieved at the same time by using symbolwise differential encoding and forming the Alamouti matrix afterwards. However, unfortunately this encoding scheme does not work. This is demonstrated as follows. Suppose that we employ symbolwise differential encoding (4.54) x2(t) x2(t 1) s(t) and use these elements to form the Alamouti matrix X(t 1) xi(t1 1) x (t1) X(t t) = X2(t ) Xt(t ) J (4.55) X( x(t) x) (t) xI(t 1)s(t) xt(t 1)s(t) X2(t) X*t) x2tl)St) x*I(t l)S*Lt) The above code matrix has the property that Ixi(t) = x2(t)l = I(t)l 2(t) 1 for all t and hence the signal envelope is constant for both transmit antennas all the time. Also X(t) is unitary for all t since it is an Alamouti matrix. The observation model (4.29) becomes (cf. Section 4.2) rY(t 1) Y(t) =H X(t1) X(t) + E(t 1) E(t)] (4.56) HX(t ) I XH(t i)X(t) + E(t 1) E(t) Employing the noncoherent detection rule (4.30 to the above model (4.56) the following detection rule is obtained max Re{ Tr. (XH(t 1)X(t)Y(t)Y (t ))j I si(t),s2(t)eS I I = max s1 (t),s2 (t)CS R (t 1) ( 1) ) st (t) (t 1) s (t) (t) 1 2(t) X(t1) 1) [2(t 1) s2(t) X (t 1). s(t) Smax s (t),s2(t)e Re Tr t 2 (t) + 2 (t) z(t l)x) (t 1) (t) S;(t)) X1 [ _( (t l)X2 (t 1) (s1(t) s2(t)) *(t)+ S* (t) 1)}} (4.57) The above maximization problem (4.57) cannot be solved unless xi(t 1) and x2(t 1) are known. For example, consider the following two sets of values {sI(t),S 2(t),t 1)X2(t 1)} {1, 1,1} (4.58) {1(t), S2(,x(t 1)X2(t 1)} {1,1, 1} (4.59) These make the metric in (4.57) equal to Re Tr 0 2 YH(tY(t 1) I 2 0 which implies that the sets {si, a2} = {1, 1} and {si, 2} = identifiable if xl(t 1) and x2(t 1) are unknown. (4.60) {1, 1 are not and 38 10 1 nt=2, Code (4.46) (basic msg. alone)  nt=2, Suboptimal detector, Code (4.46)(4.39) (basic msg.)  nt=2, Suboptimal detector, Code (4.46)(4.39) (add. msg.) 102 nt=2, JointML, Code (4.46)(4.39) (basic msg.) . nt=2, JointML, Code (4.46)(4.39) (add. msg.) I I 2 104 15 20 25 30 35 Signaltonoise ratio [dB] Figure 49: MonteCarlo Simulation of the BER performance for the suboptimal receiver (4.49) and (4.50) in Section 4.5, for the joint ML receiver and for the basic message only system with Rb = 2, R = 1 code. The curves " x " and " x " overlap. 101 10 II nt=2, Group Codes (4.46) (basic msg. alone) ' nt=2, Group Codes (4.46)(4.39) (basic msg.) . nt=2, Groups Codes (4.46)(4.39) (add. msg.) Sn=2, Alamouti Codes (4.52) (basic msg. alone) *1 n t2, Alamouti Codes (4.52)(4.39) (basic msg.) "1 nt=2, Alamouti Codes (4.52)(4.39) (add. msg.) a  10  a ) I I 104 15 20 25 30 35 40 Signaltonoise ratio [dB] Figure 410: Monte Carlo Simulation result for the BER performance of R, = 2, Ra = 1 system, with differential group codes and differential Alamouti codes. Note that the solid and dash curves with "x" marks overlap. CHAPTER 5 CONVOLUTIONAL PRECODING 5.1 Introduction to Convolutional Codes In Pursley et al. [12], convolutional coding are introduced for extra error protection before the data symbols are further encoded by the nonuniform PSK modulation. For completeness we introduce here the basic concept of convolutional coding as a tool to improve the BER performance. Convolutional coding is a kind of forward error correction (FEC) technique, which means that no feedback channel is required in contrast to, e.g. Automatic Repeat Request (ARQ). The idea of convolutional coding is simple: To improve the error perfor mance of a transmission link by adding some carefully designed redundant bits to the data before it is transmitted through the channel. This process of adding redundancy is known as channel coding. In contrast to block codes which operate on large message blocks, convolutional codes operate on a continuous stream of data bits. Convolutional coding is a FEC technique which is especially suitable to channels which are contaminated by additive white gaussian noise (AWGN). By using convolutional coding, BER performance can be improved significantly at the same SNR, at the cost of lowering the data rate. Meanwhile this rate loss can be compensated for by increasing the constellation size. There are two important parameters for a particular convolutional codes, namely the code rate and the constraint length. The code rate, k/n, is simply the ratio of the number of data bits input to the convolutional encoder (k) to the First output Input ZAt  Z1 \ // Figure 5 1: A simple rate 1/2 convolutional encoder. number of output bits by the convolutional encoder (n), can also be a measure of the efficiency of the code. The constraint length, K, denotes the "1. 1,;I !: or "memory" of the convolutional encoder, that is, each output bit is affected by K previous input bits. For years, convolutional coding has been the predominant FEC technique. But there is a tradeoff of employing convolutional coding. For example, using a 1/2 rate convolutional codes, the data rate will be deducted to half, provided that the modulation technique is the same. This is simply because with rate 1/2 convolutional encoding, two channel symbols are needed to be transmitted per one data bit. But the advantage is that, the convolutional coding can improve the error performance at the same SNR, at the cost that the data rate will decrease by a factor of n/k. Figure 5 1 shows a typical rate 1/2 convolutional encoder, built with two shift registers and two modulo2 adders. One should p ,i' attention that an interleaver is usually used to interleave the encoded data bit stream from the output of the convolutional encoder. The benefit of interleaving the data bits is that it can provide time diversity: consecutive bits are likely to be contaminated by a temporary strong interference or noise, or sustained by deep fading channel environment, this can make the error correcting algorithm at the convolutional decoder fails to correct bit errors and recover the correct information. With sufficient interleaving, the originally consecutive bits are now separated far apart as they are being transmitted through the wireless channel, so that we can reasonably assume that the bits nearby each other after deinterleaving have undergone independent fading and are affected by independent noise and interference, this is desirable to the convolutional decoder as consecutive bits are not likely to encounter errors simultaneously. Also the design task of the ML detector at the receiver is also simplified. 5.2 Convolutional Codes Applied to SpaceTime Coding System We are going to investigate the error performance of the system if a con volutional encoder is employ, l to the information bit streams prior to STBC mapping. It is expected that the BER performance can be improved substan tially by such kind of precoding in an adverse channel environment. This simply trades the error performance with data rate. To start with, we choose a simple rate1/2 convolutional encoder with constraint length K = 3, generators are 5, 7 in octal. As for our Rb = ,R, 1 code, there are two bits carried in the basic 1liv. and two bits carried in the additional 1~v r for each transmit time interval, let sl, s2 denote the two bits carried in the basic 1v. r and S3, s4 denote the twos bits carried in the additional 1 r, respectively. And now we have four separate (independent) bit streams, for example they may look like the following i 0, 1, 1,0, 1, 1,0, 1, 1, * 2 1, 0, 1, 1 0, 0,0, 1, (5.1) s3 = 0,0,0, 1,0, 1, 1, 1, 1, S4 = 0, 0, 1,0, 0, 1, 0, 1,0,... We model the sl, 82, 83, 84 as independent bit streams with random 0 and 1. For each bit stream we pass it through a standalone binary convolutional encoder stated above. After that, we would have four encoded bit streams, denoted by s s2, s3, s. Notice that the encoded streams are not of the length of the raw data streams, for the rate1/2 encoder used in our study here, the length of the encoded streams is doubled, we sacrifice the data rate by half to trade for better BER performance. The four encoded bit streams are then being passed through the STBC mapper as before and sent via the MIMO system. 5.2.1 HardDecision Decoding We get the estimated bit streams si', S2, S3', 54' at the output of the harddecision symbol detector, we then pass them through the Viterbi decoder to decode them back into the raw data streams. BER performance of such a harddecision decoding system is shown comparing to the unencoded system. 5.2.2 SoftDecision Decoding For a softdecision decoder, the input is no longer a precise 0 or 1 binary estimation but the likelihood ratio P(c = 1)/P(c = 0) of each bits. To calculate the likelihood ratio of each individual bit we have to consider the receiver structure. For the differential encoding and decoding system in section 4.2, to decode the differentially encoded information message, the receiver has to consider two consecutive received data blocks Y(t 1) Y(t) = HX(t ) I U(t) + E(t 1) E(t) (5.2) S effective channel He '' v Ye Ue Ee where Ye and Ue stand for the "effectli, received data block and the trans mitted matrix, respectively. To detect the transmitted matrix Ue given Y,, the noncoherent detection technique discussed in section 4.2 can be used. Though there is one main difference: For harddecision decoding in section 4.2 we do not need to know the exact value of the likelihood function, but simply choose the Ue to maximize it. For this purpose we only need to concentrate the func tion against the unknown channel, and simply treat the noise variance a2 as an unknown constant would be good enough. However for softdecision decoding we have to not only maximize the likelihood function against U,, but have to evaluate the exact values of the likelihood function for any possible transmitted matrix Ue, to achieve this we have to concentrate the likelihood function against both the unknown channel H and the unknown noise variance a2. The logarithm of the likelihood function is the following [16] L(YIU,, He, a2)= 2Nnr log a2 _ 2Tr ((Y HeUe)(Y HeUe)H} (5.3) Maximization of (5.3) with respect to a2 requires that 9L(Y I U, H, a2) 0 a72 (5.4) 1 1 S2Nnr + Tr {(Y HeUe)(Y HeUe)H} 0 a2 a4 which yields a positive solution for a2 a2 = tTr ((Y HU,)(Ye HU,)H} (5.5) 2Nn, Inserting (5.5) into (5.3) gives L1(Y U, H) = log Tr {(Y H,U)(Y, HU,)H} (5.6) We next continue to concentrate the likelihood function to eliminate He. Maximizing (5.6) with respect to He, we get He YUH(U H)U 1 tYUH (5.7) 2 where we used the fact that UUH = 21. Insertion of (5.7) into (5.6) gives the concentrated likelihood function L(YIU,) logTr (Y, YuUU) (e eUH )H (5.8) logTr YYY HYe eH eYH For a particular information bit, , sl, the likelihood ratio can then be approximated by P(si 1) UZ,:I exp (L(Ye Ue)) P(si 0) ugs:, oexp (L(Y, U )) (5.9) S:Ucs81 Tr{yyY, Uyu0uyY} 1 1Ugsi 0 Tr{ Y0YH Y0Iuuy} We can thus evaluate this likelihood ratio for each sl, s2, 83, S4 bits for every time intervals and use this soft information bit estimation as the input of the softdecision Viterbi decoder. Alternatively, the receiver may use the previously received data blocks to estimate the noise variance. Doing so we just need to insert the estimated noise variance into the likelihood function, and it is not necessary to concentrate the likelihood function with respect to the noise variance. If we estimate the noise variance for a long enough time and take the average value, we will hopefully get a very accurate estimation. To get the estimated P2 from the received data blocks we simply need to refer back to (5.5) 62 = Tr Ye H,U,)(Y, E H,U,)H} (5.10) 2Nnr Substituting (5.7) into (5.10) gives 1 1 H 1 H 2 Tr (Y YIU UU)(Y _YU U) H 2Nnr 2 2 (5.) 1 r H H(5.11) 1 Tr YYH y Y_ U U T \Y 2Nn, 2ee H Substituting the 62 and He = YUU back into (5.3) yields the likelihood function with estimated noise variance 2) Tr l _tH tY'U( H( vy L(YIU,,2) 2Nrnlog 2 Tr 1Y H UY (5.12) With an accurate c2 we may expect that this likelihood function performs better than the concentrated function with unknown noise variance since there is one less unknown variable. Figure 52 shows the simulation results of the performance of both harddecision and softdecision schemes. U ^ 0% 8 . \ ^ v " \\5 \\ \\ %%, Y3 %.% % SBasic msg. (hard decision) Add. msg. (hard decision) ) Basic msg. (soft decision with concentrated ML function) Add. msg. (soft decision with concentrated M L function) Basic msg. (soft decision with exactly known noise variance) Add. msg. (soft decision with exactly known noise variance) )Basic msg. (soft decision with est. noise variance) Add. msg. (soft decision with est. noise variance) , S  X\ 'Xx 100 101 102 I O 2 0 103 I  104 0 111111 , 8 10 12 14 Signaltonoise ratio [dB] 16 18 Figure 52: With rate1/2 convolutional coding, comparison of BER performance are shown for harddecision detection, softdecision detection with concentrated likelihood function, and softdecision with estimated noise variance. '.0~ C> 105 4 I \N \\ CHAPTER 6 A NETWORK APPLICATION EXAMPLE We consider employing our nonuniform spacetime codes in a broadcasting telecommunication system. First we present a short introduction to contempo rary typical wireless cellular systems. The concept of a wireless cellular system is fairly simple: a large region to be served are divided into many small areas called "cells". Each cell is serviced by one or more base stations) located at the center or the corners of the cell. The advantage of such a cellular system is that the available frequencies band can be reused in cells which are far apart; thus the same available frequency band can now be enjoi, l by a small area which may consist of several small cells. Figure 61 shows a 7cell frequency reuse scheme. A cluster is formed by seven cells. A cluster can use all the available bandwidth, and the whole frequency band is being reused in all other clusters. So the whole frequency band is now shared by seven cells only, thus each user in the cluster should be able to get a adequate amount of usable bandwidth. Since the .,.i i:ent clusters use exactly the same frequency band, a power control scheme can be employ, l to keep the SignaltoInterferenceRatio (SIR) under certain threshold. Now we are moving on to see what is happening on one particular cell. Figure 62 shows a typical cell. For simplicity we assume the shape of the cell to be circular, and that the base station is located at the center. Let Ro represents the cell radius. Typically the SNR at the cell border should be the lowest. To predict the signal strength at various places of the cell, some sort of propagation model has to be used. Here we first introduce the free space propagation model Figure 6 1: 7cell frequency reuse system. due to its simplicity. Assuming there exists a LineOfSight (LOS) propagation path, then the received signal strength can be predicted by PtGtG,A 2 Pr(d) (= 4 2 (6.1) (4w)2d2 where Pt is the transmit power at the transmit antenna, Gt and G, are the transmit and receive antenna gains respectively, A is the wavelength of the transmitted signal, d is the propagation distance. In the previous discussion of spacetime coding it is assumed that we have a frequ, ii' flat channel, this means that we must use a narrowband modulated signal. It is well known that the received signal strength attenuates as d2 in free space theoretically, but in practical environment for a wireless link, the LOS path usually not exists and the propagation may encounter various loss such as scattering and other obstructions, so the attenuation is usually much higher. Even with a LOS path, the received signal strength usually attenuates faster than d2 due to destructive reflection. Another commonly used model is the Ra RbRo Figure 62: Typical cell layout for a wireless telecommunication system. i, ./:, ir,.. : path loss model P (d) = Po d (6.2) do Here Po is the received power at a reference point, do is the distance between the transmitter and the reference point, and n is the path loss exponent. For ideal free space, n = 2. For an urban area we may take n 3 or 4 [24]. Due to the unpredictability of the radio environment the above model can only represent the average received power. In fact the actual received power fluctuates vastly around the average value since the propagation environment varies at different time/space (varying reflective objects, obstructions, weather, etc). From a system design point of view the exact level of received power is quite hard to predict and this sort of power fluctuation may be modelled as a random quantity in a mathematical framework. So the Lognormal Fading model is being introduced lOloglo rP(d) [dB] = lOloglo Po [dB] 10 log( ) + X, (6.3) do 51 where X, is a zeromean Gaussian distributed random variable with variance O2 (Typical w 6 dB over large/small scale fading). Thus we may w that for any point at the cell, the instantaneous SNR is a Gaussian random variable with the mean value given by the Log Distance Path Loss Model. SNR SNR + X, (6.4) Since now the SNR at any point is modelled as a random variable then the instantaneous BER performance is also random, we can define the coverage area of a cell. Define the coverage radius to be the radius of the area such that a specified detection performance (in terms of BER) is achievable with a certain probability that we call the Quality of Service (QoS). For example, if we let Ro denote the 9' coverage radius then within the area of radius Ro the message can be detected at a specified BER during 9' '. the of total time. If nonuniform spacetime codes are used to increase the data rate for the more capable receivers, it is clear that the coverage radius for the transmission of the basic message under the same QoS will decrease to Rb < R0. Meanwhile the coverage radius of the transmission of the additional message now becomes Ra; of course its value depends on QoS for the additional message. For the simplicity of our analysis here let us assume the same QoS for both the basic and the additional message. Denote the "tolerable" performance loss for the detection of the basic message as A dB. Assuming logdistance fading it can be shown that R 0o (6.5) Ro 52 R2 r2 R 10 Ti (6.6) "o where F is the performance difference (in dB) between the detection of the additional message and that of the standard uniform constellation system. The parameter n is the path loss exponent for the radio link environment (i.e. Power oc (distance)"; for details see Rappaport et al. [24]). The above equations are also valid in a lognormal fading model. It is not difficult to obtain the result above: Denote the SNR required to have a certain BER performance as SNRth[dB]. Then, for example, the 9''. coverage radius Ro can be determined as it satisfies P [Pr\(Ro)[dB] + Xsigma > SNRth[dB] 9= (6.7) When we apply the nonuniform modulation the BER performance of the basic message deteriorates by A dB, which means that it requires a A dB higher SNR to obtain the same performance, so P P(Rb)dB] + Xsigma > SNRth[dB] + A] = 9' I (6.8) Comparing (6.7) and (6.8) we get Pr(Rb)[dB] P,(Ro)[dB] + A R. ((6.9) tlOloglo Po [dB] 10n log( ) lOlog Po [dB] + A(6.9) RO and (6.5) follows immediately, (6.6) can be shown similarly. Figure 63 shows how Rb and Ra vary with A for the Rb = Ra 1 system, with the power attenuation factor of the radio link taken to be n = 2 and n = 4; these are typical values for freespace and downtown areas, respectively [24]. It can be seen that employing the nonuniform constellation does not increase the total throughput of the system (i.e., the sum of the data rate for all users) in the case n = 2. For example, if the coverage area for the basic message 53 (0 1 Illl ]  i ii l e Basic msg. (n=2) =0.9e N Add. msg. (n=2) o R =2 basic msg. only (n=2) S0.8 o Basic msg. (n=4) S, x Add. msg. (n=4) S_ R =2 basic msg. only (n=4) o 0.7 b 0.6   ^ ) 0.5 N E 0.4 x  o *** 0 X* 0.3  c0.2  S0.1 0 I 0 0.5 1 1.5 2 2.5 3 3.5 4 Performance loss of basic msg. (dB) Figure 63: Coverage area for the basic/additional messages as a function of the "tolerable" performance loss, for the Rb 1, R = 1 code. The horizontal curves represent the relative coverage area for the Rb = 2 code with a basic message only. is allowed to decrease to 0.8 times of that of the original cell area, the coverage area of the additional message can only reach 0.15 times of that of the original cell area, which results in a decrease in overall system throughput. However for the case n = 4, the use of a nonuniform code increases the total throughput. This observation is not hard to understand; for n = 4 the area closely surrounding the base station enjoys a relatively much higher SNR than the area close to the border of the cell and hence the coverage area for the additional can be extended further. Note that the coverage areas of the nonuniform lF ,v. 1, code (4.38) and 0. 0. En o Z0 cu a), 0 0( O II" 8 C ^I 0. e SISO basic msg. (n=2) 45  SISO add. msg. (n=2) Rate2 SISO basic msg. only (n=2) o SISO basic msg. (n=4)  SISO add. msg. (n=4) Rate2 SISO basic msg. only (n=4) 35  25   Jc 25 ~~  .2 . lb  M 05  n1 ......,_. 0 0.5 1 1.5 2 2 Performance loss of SISO basic msg. (dB) Figure 64: Coverage area for the SISO basic/additional messages as a function of the "tolerable" performance loss. The horizontal curves represent the relative coverage area for the SISO rate2 code with a basic message only. (4.39) (with Rb Ra = 1) tend to approach that of the Rb = 2 code (4.46) when A increases. The performance advantage of a 2TX system upon 1TX system is illus trated in Figure 64. The coverage area of the 1TX system is only about 0.2 times as that of a 2TX system for n = 4. 0. 0 0 0 CHAPTER 7 CONCLUDING REMARKS AND FUTURE WORKS We have presented new nonuniform spacetime codes that can be encoded and detected differentially, and that are based entirely on phaseshift keying. We also discussed analytical criteria for code construction and optimization, and we compared its performance with that of a scheme based on the Alamouti code. We also studied a suboptimal detector and its performance, which we found to be satisfactory (a loss within 0.3 dB compared to joint ML decoding). We also demonstrated how nonuniform spacetime codes in a broadcasting telecommunication system can increase the total throughput. It may be argued that using already established nonuniform constellations for single transmit antenna systems together with, for instance, known linear spacetime codes should be a natural approach to the problem of designing nonuniform spacetime constellations. However, there are at least two problems associated with such an approach. First, it may in general not be optimal, simply because we are optimizing over the class of spacetime codes and the class of nonuniform singleantenna constellations separately, instead of optimizing over the class of nonuniform MIMO constellations. Second, we found it difficult to incorporate constraints (such as constant envelope after differential encoding), that are desired or required from a practical implementation point of view. Therefore, we believe that it may be advantageous to design nonuniform space time constellations by approaching the problem from first principles. In this thesis, the nonuniform spacetime block codes are chosen in quite an ,il I trary" way, and they are optimized with the help of chernoff bounds of BER, which are ahv, loose. We can expect the ..p 'l ii',!" nonuniform group codes , ii. 1. in this paper are indeed not optimal. Future works may include 1. Calculation of exact theoretical BER 2. 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